Overall Reaction Rate Study of Thermal Methane Cracking in NonIsothermal Conditions
Mahdi Yousefi ^{1,*}, Scott Donne ^{1}, Shabnam Bahremand Abrasi ^{2}, Mohammad Yousefi ^{3}

PRC for Frontier Energy Technologies and Utilization, University of Newcastle, Callaghan, NSW 2308, Australia

Environmental Remediation (GCER), University of Newcastle, Callaghan, NSW 2308, Australia

Tehran Azad University of University Science and Research, Chemical Engineering Department, Ferdos Blvd, Tehran, Iran
* Correspondence: Mahdi Yousefi
Academic Editors: Zhiwei Ma and Huashan Bao
Special Issue: Decarbonisation of Heating and Cooling
Received: April 10, 2023  Accepted: September 07, 2023  Published: September 13, 2023
Journal of Energy and Power Technology 2023, Volume 5, Issue 3, doi:10.21926/jept.2303028
Recommended citation: Yousefi M, Donne S, Bahremand Abrasi S, Yousefi M. Overall Reaction Rate Study of Thermal Methane Cracking in NonIsothermal Conditions. Journal of Energy and Power Technology 2023; 5(3): 028; doi:10.21926/jept.2303028.
© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.
Abstract
The thermal cracking of methane (TMC) is a significant reaction occurring above 850°C, which proceeds in two stages: nonisothermally and isothermally. However, most existing studies have focused on obtaining reaction rates under isothermal conditions [1], limiting their applicability to practical industrial reactor conditions. This novel research aims to determine the overall thermal decomposition rate of methane to hydrogen and carbon in adiabatic conditions, covering the range of unstable industrial reactor temperatures (850 to 1200°C). The Coats and Redfern modelfitting method was employed to calculate the reaction rate under nonisothermal conditions, and the resulting models were compared with experimental data. The findings reveal the Contracting Cylinder model as the bestfit mathematical representation with less than ±2.8% error. By extending the kinetic model to nonisothermal conditions, this approach addresses a critical aspect of realworld applications.
Keywords
Thermal methane cracking; hydrogen production; overall reaction rate; adiabatic conditions
1. Introduction
Thermal cracking of natural gas is a promising process for simultaneous hydrogen production and distribution [1]. In this process, pure hydrogen is obtained from the thermal decomposition of natural gas. The availability of natural gas in many regions, as well as the possibility of producing very pure hydrogen without advanced purification systems, make it an attractive option. It could eliminate the need for extensive hydrogen transportation and storage, which are major obstacles to developing hydrogen applications [1,2,3].
However, thermal methane cracking is still an immature technology for industrial and semiindustrial scales, and more research is needed to understand all the advantages and challenges. Among these challenges is obtaining an accurate examination of the overall reaction rate at all temperature intervals. Despite numerous investigations, an exact characterisation of the cracking reaction has not been achieved [4]. Although methane is the simplest hydrocarbon, the mechanisms of its decomposition reactions are not fully understood, and precise characterisation of the mechanisms has not been reported [5,6,7]. In their study, Mahdi et al. [4] provided an overview of the subreactions involved in the thermal methane cracking process (Figure 1) and the different species that contribute to the cracking reactions from the reactant CH_{4} to the final products (H_{2} and C).
Figure 1 Thermal methane cracking kinetic model shows the intermediates products in MTC immediately converted to hydrogen and carbon and are on the path from the reactant (methane) to the final products [4].
Previous studies have mainly focused on the mechanism of intermediate reactions and materials, leading to general mass transfer coefficient (MTC) reaction rates that are only valid for isothermal conditions. For example, Wullenkord et al. [8] tried to investigate the conversion rate of methane at 1200 to 1600°C. It should be noted that the thermal decomposition reaction of methane above 850°C is significant and cannot be ignored and should be considered when methane is heated up in a nonisothermal condition. It is also wellconfirmed with other published studies, such as Lee et al. [9] and Trommer et al. [10] conducted for temperatures lower than 850°C. Holmen et al. [7] described a model for the production and consumption of the main hydrocarbons at both high and low conversion rates of CH_{4} in the temperature range of 11961450°C, and the results of the experimental tests and the model predictions were almost close. However, it could not cover lower temperature rates as the method was based on an isothermal condition. The nature of most industrial reactors is such that the gas temperature varies throughout the reactor, and the reaction occurs when the feed gas is warming up inside the reactor. Thus, it is necessary to obtain an overall reaction rate covering both nonisothermal and isothermal stages to model an unstable reactor. This research aims to study the overall reaction rate of thermal decomposition of methane in the 850 to 1200°C temperature range in adiabatic conditions using a modelfitting method and to compare the results with experimental data obtained using COMSOL Multiphysics software [4].
2. Background
This study is based on experimental data obtained by the authors and recently published in the Frontiers in Energy Research Journal (2022) [4]. The study aimed to convert methane into hydrogen and methane using a tubular reactor filled with inert ceramic packings under isothermal conditions in the temperature range of 850 to 1200°C. According to Table 1, which shows the concentration of hydrogen at 850 to 1170°C obtained in the experiments, the reaction rate at temperatures lower than 850°C is very slow, indicating negligible conversion of methane during the heating up process. However, at higher temperatures, the cracking reaction is significant, and the reaction can be divided into isothermal and nonisothermal steps as the temperature of the methane gas increases.
Table 1 the concentration of hydrogen at the reactor outlet at 850 to 1170°C and 0.2 to 2 l/min methane feed rate [4].
2.1 NonIsothermal Reaction Rate Models Description
The nonisothermal reaction rate model can be described using Equation (1), which gives the overall reaction rate as a function of time, temperature, and the temperaturedependent kinetic constant. The degree of conversion at any time is calculated using Equation (2). Equation (3) provides a way to define Equation (1) in terms of the reaction rate constant and the integral model of the reaction. Table 2 lists some of the common mathematical integral models used for nonisothermal reaction rates, based on different f(α) and g(α).
Table 2 The common mathematical integral models for nonisothermal reaction rates models [11,12].
Under nonisothermal reaction conditions, the kinetic constant k(T) can be described by the Arnious equation. Equations (4) and (5) explain Equations (1) and (3), respectively, in terms of the kinetic constant and the heating rate (β = dT/dt). Equation (6) is the integration of Equations (4) and (5), and it can be rewritten as Equation (7) by considering x = E/RT, where L(x) is an exponential integral that does not have an analytical solution but can be estimated numerically.
There are two main nonisothermal approaches to obtaining the kinetic parameters: modelfitting and modelfree methods. The modelfitting method involves fitting different models to α vs temperature curves and defining E and A parameters concurrently. The Coats and Redfern method is one of the most common models for modelfitting methods and uses an asymptotic series expansion, as shown in Equation (9), where $T^m$ is the mean temperature.
Equation (1) is a mathematical model to describe overall nonisothermal reaction rates:
\[ \frac{d\alpha}{dt}=k(T)\times f(\alpha) \tag{1} \]
\[ \alpha=\frac{m_0m_t}{m_0m_f} \tag{2} \]
α is the degree of conversion at any time calculated by equation (2), t is the time, T is temperature, k(T) is the temperaturedependent kinetic constant, f(α) is the reaction model, which is an empirical function to express the change in the reactivity as the reactions proceed (dimensionless). m_{0} is the initial mass of the sample and m_{t} is the timedependent mass. m_{f} is the final mass of the sample.
Equation (1), which describes the overall nonisothermal reaction rate (dα/dt), can be related to Equation (3) as follows: g(α) = kt. In this relationship, g(α) represents the integral model of the reaction, and k is the reaction rate constant. Equation (3) provides a simplified representation of the overall reaction rate, expressing it in terms of the reaction rate constant (k) and the integral model of the reaction (g(α)), obtained by integrating the overall reaction rate with respect to time.
\[ g(\alpha)=kt \tag{3} \]
Where k is the reaction rate constant, and g(α) is the integral model of the reaction. Table 2 lists some important reaction models based on different f(α) and g(α). Under nonisothermal reaction conditions and describing the constant kinetic k(T) by Arnious equation once equations (1) and (3) are explained based on equations (4) and (5):
\[ \frac{d\alpha}{dt}=\frac{A}{\beta}\times e^{\frac{E}{RT}}\times F(\alpha) \tag{4} \]
\[ g(\alpha)=A\times e^{\frac{E}{RT}}\times t \tag{5} \]
where β is the heating rate (β = dT/dt). Equation (6) is an integration of equations (4) and (5):
\[ g(\alpha)=\frac A\beta\int_{0}^{T}e^{\frac{E}{RT}}dT \tag{6} \]
If considering the x = E/RT once the equation (6) becomes:
\[ g(\alpha)=\frac{AE}{RT}\int_{0}^{\infty}\frac{e^{x}}{x^{2}}dx \tag{7} \]
It can be rewritten:
\[ g(\alpha)=\frac{AE}{RT}L(x) \tag{8} \]
Where L(x) is an exponential integral and does not have an analytical solution, but there are some numerical estimates. The modelfitting and modelfree methods are the main nonisothermal approaches to obtaining the kinetic parameters. The modelfitting method includes fitting different models to α vs temperature curves and concurrently defining E and A parameters. Coats and Redfern method is one of the most common models for modelfitting methods [11], and it uses asymptotic series expansion, which is explained by equation (9):
\[ \ln\frac{g(\alpha)}{T^2}=\ln\left[\frac{AR}{\beta E}\Big(1\frac{2RT^m}E\Big)\right]\frac E{RT} \tag{9} \]
Where, $T^m$ is mean temperature.
3. Results
The experimental data given in Table 1 were used to obtain the values of E and A corresponding to Equation (9) by plotting ln g(α)/T² versus 1/T and calculating the slope and intercept. Figure 2 shows the ln g(α)/T² versus 1/T for different reaction modelfitting, which are listed in Table 2. Table 3 provides the values of E and A for each reaction based on the different reactions obtained from Figure 2.
Figure 2 $\ln\frac{\mathrm{g}(\alpha)}{\mathrm{T}^2}$ vc 1/T for different reaction modelfitting at 900 to 1150°C and atmospheric pressure.
Table 3 The value of E and A for each reaction based on different reaction fitting at 900 to 1150°C and atmospheric pressure.
The accuracy of each model (R²) was obtained by comparing the predictable results of each model (using data in Table 3) with the experimental results obtained in tests. Figure 3 compares the ln g(α)/T² value for each model listed in Table 3 with experimental data.
Figure 3 $\ln\frac{\mathrm{g}(\alpha)}{\mathrm{T}^2}$ Versus 1/T by the data obtained in the experiments and the calculated data in Table 3 at 900 to 1150°C for different reaction modelfitting.
Table 4 provides the average error and actual error for the rate models at different temperatures. The lowest error was observed for the contracting cylinder model with about 2.8%, followed by diffusion control (Janders) with 4.95% and diffusion control (Crank) with 5.03%.
Table 4 The average error and the actual error for the rate models at different temperatures.
Using the contracting cylinder model, Equation 9 was rewritten as Equation 10, which can be used for a temperature range of 900 to 1170°C with a high approximation.
With the help of COMSOL Multiphysics software, the experimental reactor used in this research was modeled using Equation 10. The average error in this evaluation was less than 3%, demonstrating that the overall reaction rate was governed with acceptable accuracy. Figure 4 provides a comparison of the CFD model results with the experimental results obtained at different temperatures and feed rates up to 2.5 lit/min.
Figure 4 Comparison of CFD model results with experimental results obtained at (a) 850°C, (b) 950°C, (c) 1050°C and (d) 1150°C and feed rates up to 2.5 lit/min.
Using the experimental data given in Table 2, the values of E and A corresponding to equation 9 are obtained by plotting $ln\frac{g(\alpha)}{T^2}$ Versus 1/T the (slope and intercept). Figure 2 indicates the $ln\frac{g(\alpha)}{T^2}$ vs 1/T for different reaction modelfitting, which are listed in Table 2.
Table 3 indicates the value of E and A for each reaction based on the different reactions obtained from Figure 2.
The accuracy of each model (R²) is obtained by comparing the predictable results of each model (using data in Table 3) with the experimental results obtained in tests. Figure 3 compares the $ln\frac{g(\alpha)}{T^2}$ value for each model listed in Table 3 with experimental data.
The average and actual errors at different temperatures for each model have been given in Table 4. As seen, the lowest error is for the contracting cylinder model with about 2.8%, followed by diffusion control (Janders) with 4.95% and diffusion control (Crank) with 5.03%.
By selecting the Contracting cylinder model, Equation (9) can be rewritten as the following equation, which can be used for a temperature range of 900 to 1170°C with a very high approximation.
\[ \ln\frac{2(1\alpha)^{0.5}}{T^{2}}=\ln\left[\frac{0.094}\beta\right]\frac{10835.9}T \tag{10} \]
Using Equation (10), the condition of the experimental reactor (used in this research) is modelled with the help of COMSOL Multiphysics software to compare and have a more indepth study.
In this model the test reactor dimention is modeled for simplification in explaining the physical process, mass transfer, heat and momentum. The geometry of the reactor is a cylinder that is symmetrical in terms of geometry and operating conditions, to reduce the volume and time of calculations, the reactor is simulated in two symmetrical dimensions. The thermal cracking reaction takes place in the gas phase, and the solid carbon is deposited on the ceramics. The reactor voidage changes depending on the amount of produced carbon. An ODE^{1} physics is coupled to see changes in voidage due to carbon production as a function of time.
The average error in this evaluation is less than 3%. It confidently shows that the overall reaction rate is governed with acceptable accuracy.
4. Discussion
Most previous research focused on the mechanism of intermediate reactions and materials. For instants, Holmen et al. [13] described a model for the production and consumption of the main hydrocarbons at both high and low conversion rates of CH_{4} in the temperature range of 11961450°C. All of them led to some general MTC reaction rates. The point here is that they can mostly not be used in an unstable reactor because the rate calculations are valid to use in an isothermal condition. While the nature of most industrial reactors is that the gas temperature varies throughout the reactor, and the reaction occurs when the feed gas is warming up inside the reactor. The point is that many researchers have not paid much attention to it, which led to the obtained rates being only used in a limited temperature range. For example, Wullenkord et al. [14] tried to investigate the conversion rate of methane at 1200 to 1600°C, and Kevorkian et al. [15] investigated the conversion rate of methane at 1383 to 1692°C.
Notably, the thermal decomposition reaction of methane above 850°C is significant, cannot be ignored, and should be considered in a nonisothermal condition. A more detailed study of the conditions governing the reactor by CFD modelling reveals that the reaction rate will change when methane gas reaches the reactor temperature. Afterwards, the reaction is conducted isothermally. Since the reaction rate at the beginning of the reactor and before the methane temperature reached the reactor temperature was much higher than the next stage (isothermal), it is necessary to obtain the overall reaction rate covering both stages.
A detailed analysis of the results obtained from the investigation of thermal methane cracking (TMC) under nonisothermal conditions. To achieve the objective, various kinetic models, including the Contracting Cylinder model, Coats and Redfern model, and others, were used to calculate the reaction rate in nonisothermal conditions. Each model's predictions were compared with the corresponding experimental data, enabling a comprehensive analysis of their performance.
The Contracting Cylinder model exhibited the best fit to the experimental data, with a remarkable error margin of less than ±2.8%. This result demonstrates the model's superior ability to capture the complex behavior of the methane thermal cracking reaction under nonisothermal conditions. In contrast, the other models displayed larger deviations from the experimental data, highlighting their limitations in accurately describing the kinetics of the reaction in industrial reactor settings. The comprehensive analysis of these models has provided valuable insights into their applicability and the significance of considering nonisothermal conditions. The comparison of model predictions with experimental data holds paramount importance as it validates the accuracy and reliability of each kinetic model under realworld conditions. Using nonisothermal models addresses a crucial aspect of the practical application of thermal methane cracking for hydrogen production.
The findings allow us to confidently assess the most suitable kinetic model for industrialscale reactors, where nonisothermal temperature profiles are the norm. The Contracting Cylinder model's success in predicting the conversion rates of methane to hydrogen and carbon in a nonisothermal environment signifies its potential to enhance reactor design and optimise hydrogen production efficiency.
In conclusion, the comprehensive analysis presented in this section highlights the significance of adopting nonisothermal models to accurately represent the behavior of thermal methane cracking under industrial reactor conditions. The superiority of the Contracting Cylinder model underscores its potential for practical application and advances our understanding of the complex kinetics involved in this process. This study bridges the knowledge gap and paves the way for efficient hydrogen production in industrial and semiindustrial settings.
5. Conclusions
In conclusion, this research aimed to obtain the overall reaction rate of thermal methane cracking (TMC) under nonisothermal conditions covering the temperature range of 8501200°C. The Coats and Redfern method was used to calculate the reaction rate in nonisothermal conditions, and the Contracting Cylinder model was found to be the most accurate with less than ±2.8% error. This study provides a valuable contribution to the understanding of TMC under nonisothermal conditions and lays the foundation for developing more efficient industrial reactors for methane cracking.
Acknowledgments
The authors would like to acknowledge the financial support provided by the University of Newcastle, Australia, for the work presented in this paper.
Author Contributions
Dr Mahdi Yousefi played a pivotal role in the research process and served as the primary author responsible for drafting the paper. He conducted the research within the framework of the project titled "A Novel Fuel Converter for the Production of Hydrogen from Natural Gas" at the University of Newcastle (UON). Professor Scott Donne made significant contributions as the principal supervisor of the research, providing valuable guidance and expertise throughout the study. Dr Shabnam Bahremand Abrasi contributed to the research by handling data analysis, formatting, and proofreading tasks, ensuring the accuracy and clarity of the manuscript. Engineer Mohammad Yousefi also played a crucial role in this research, particularly in assisting with the computational fluid dynamics (CFD) modelling, which was essential for the study's methodology. Collectively, these contributions from the authors ensured the successful completion of the research project and the preparation of this manuscript.
Competing Interests
The authors have declared that no competing interests exist.
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